September  2011, 3(3): 313-322. doi: 10.3934/jgm.2011.3.313

Euler equations on a semi-direct product of the diffeomorphisms group by itself

1. 

Institute for Applied Mathematics, University of Hanover, D-30167 Hanover, Germany

2. 

School of Mathematical Science, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland

3. 

LATP, CNRS & University of Provence, 39 Rue F. Joliot-Curie, 13453 Marseille Cedex 13

Received  August 2011 Revised  September 2011 Published  November 2011

The geodesic equations of a class of right invariant metrics on the semi-direct product $Diff(\mathbb{S}^1)$Ⓢ$Diff(\mathbb{S}^1)$ are studied. The equations are explicitly described, they have the form of a system of coupled equations of Camassa-Holm type and possess singular (peakon) solutions. Their integrability is further investigated, however no compatible bi-Hamiltonian structures on the corresponding dual Lie algebra $(Vect(\mathbb{S}^1)$Ⓢ$Vect(\mathbb{S}^1))^{*}$ are found.
Citation: Joachim Escher, Rossen Ivanov, Boris Kolev. Euler equations on a semi-direct product of the diffeomorphisms group by itself. Journal of Geometric Mechanics, 2011, 3 (3) : 313-322. doi: 10.3934/jgm.2011.3.313
References:
[1]

V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233.  Google Scholar

[2]

V. I. Arnold and B. Khesin, "Topological Methods in Hydrodynamics," Applied Mathematical Sciences, 125, Springer-Verlag, New York, 1998.  Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singapore), 5 (2007), 1-27. doi: 10.1142/S0219530507000857.  Google Scholar

[4]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.  Google Scholar

[5]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[6]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757.  Google Scholar

[7]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.  Google Scholar

[8]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.  Google Scholar

[9]

A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050.  Google Scholar

[10]

A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35 (2002), R51-R79. doi: 10.1088/0305-4470/35/32/201.  Google Scholar

[11]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.  Google Scholar

[12]

A. Constantin and B. Kolev, Integrability of invariant metrics on the diffeomorphism group of the circle, J. Nonlinear Sci., 16 (2006), 109-122. doi: 10.1007/s00332-005-0707-4.  Google Scholar

[13]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar

[14]

C. Cotter, D. Holm, R. I. Ivanov and J. R. Percival, Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation, J. Phys. A: Math. Theor., 44 (2011), 265-205. doi: 10.1088/1751-8113/44/26/265205.  Google Scholar

[15]

J. Escher, Non-metric two-component Euler equations on the circle, Monatshefte für Mathematik, published online, 2011. doi: 10.1007/s00605-011-0323-3.  Google Scholar

[16]

I. M. Gel'fand and D. B. Fuks, Cohomologies of the Lie algebra of vector fields on the circle, Funkcional. Anal. i Priložen, 2 (1968), 92-93.  Google Scholar

[17]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222.  Google Scholar

[18]

D. J. Henry, Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12 (2005), 342-347. doi: 10.2991/jnmp.2005.12.3.3.  Google Scholar

[19]

D. D. Holm and R. I. Ivanov, Smooth and peaked solitons of the Camassa-Holm equation and applications, J. of Geometry and Symmetry in Physics, 22 (2011), 13-49. Google Scholar

[20]

D. D. Holm and J. E.Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation, in "The Breadth of Symplectic and Poisson Geometry" (eds. J. E. Marsden and T. S. Ratiu), Progr. Math., 232, Birkhäuser Boston, Boston, MA, (2005), 203-235.  Google Scholar

[21]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.  Google Scholar

[22]

R. I. Ivanov, Water waves and integrability, Philos. Trans. Roy. Soc. Lond. Ser. A. Math. Phys. Eng. Sci., 365 (2007), 2267-2280. doi: 10.1098/rsta.2007.2007.  Google Scholar

[23]

R. I. Ivanov, Two-component integrable systems modelling shallow water waves: The constant vorticity case, Wave Motion, 46 (2009), 389-396. doi: 10.1016/j.wavemoti.2009.06.012.  Google Scholar

[24]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and Related Models for Water Waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224.  Google Scholar

[25]

B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4.  Google Scholar

[26]

B. Kolev, Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2333-2357.  Google Scholar

[27]

G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.  Google Scholar

[28]

I. Vaisman, "Lectures on the Geometry of Poisson Manifolds," Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994.  Google Scholar

[29]

P. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system,, Int. Math. Res. Not. IMRN, 2010 (): 1981.   Google Scholar

show all references

References:
[1]

V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233.  Google Scholar

[2]

V. I. Arnold and B. Khesin, "Topological Methods in Hydrodynamics," Applied Mathematical Sciences, 125, Springer-Verlag, New York, 1998.  Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singapore), 5 (2007), 1-27. doi: 10.1142/S0219530507000857.  Google Scholar

[4]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.  Google Scholar

[5]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[6]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757.  Google Scholar

[7]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.  Google Scholar

[8]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.  Google Scholar

[9]

A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050.  Google Scholar

[10]

A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35 (2002), R51-R79. doi: 10.1088/0305-4470/35/32/201.  Google Scholar

[11]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.  Google Scholar

[12]

A. Constantin and B. Kolev, Integrability of invariant metrics on the diffeomorphism group of the circle, J. Nonlinear Sci., 16 (2006), 109-122. doi: 10.1007/s00332-005-0707-4.  Google Scholar

[13]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar

[14]

C. Cotter, D. Holm, R. I. Ivanov and J. R. Percival, Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation, J. Phys. A: Math. Theor., 44 (2011), 265-205. doi: 10.1088/1751-8113/44/26/265205.  Google Scholar

[15]

J. Escher, Non-metric two-component Euler equations on the circle, Monatshefte für Mathematik, published online, 2011. doi: 10.1007/s00605-011-0323-3.  Google Scholar

[16]

I. M. Gel'fand and D. B. Fuks, Cohomologies of the Lie algebra of vector fields on the circle, Funkcional. Anal. i Priložen, 2 (1968), 92-93.  Google Scholar

[17]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222.  Google Scholar

[18]

D. J. Henry, Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12 (2005), 342-347. doi: 10.2991/jnmp.2005.12.3.3.  Google Scholar

[19]

D. D. Holm and R. I. Ivanov, Smooth and peaked solitons of the Camassa-Holm equation and applications, J. of Geometry and Symmetry in Physics, 22 (2011), 13-49. Google Scholar

[20]

D. D. Holm and J. E.Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation, in "The Breadth of Symplectic and Poisson Geometry" (eds. J. E. Marsden and T. S. Ratiu), Progr. Math., 232, Birkhäuser Boston, Boston, MA, (2005), 203-235.  Google Scholar

[21]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.  Google Scholar

[22]

R. I. Ivanov, Water waves and integrability, Philos. Trans. Roy. Soc. Lond. Ser. A. Math. Phys. Eng. Sci., 365 (2007), 2267-2280. doi: 10.1098/rsta.2007.2007.  Google Scholar

[23]

R. I. Ivanov, Two-component integrable systems modelling shallow water waves: The constant vorticity case, Wave Motion, 46 (2009), 389-396. doi: 10.1016/j.wavemoti.2009.06.012.  Google Scholar

[24]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and Related Models for Water Waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224.  Google Scholar

[25]

B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4.  Google Scholar

[26]

B. Kolev, Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2333-2357.  Google Scholar

[27]

G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.  Google Scholar

[28]

I. Vaisman, "Lectures on the Geometry of Poisson Manifolds," Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994.  Google Scholar

[29]

P. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system,, Int. Math. Res. Not. IMRN, 2010 (): 1981.   Google Scholar

[1]

Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. Journal of Geometric Mechanics, 2014, 6 (3) : 335-372. doi: 10.3934/jgm.2014.6.335

[2]

Hicham Zmarrou, Ale Jan Homburg. Dynamics and bifurcations of random circle diffeomorphism. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 719-731. doi: 10.3934/dcdsb.2008.10.719

[3]

Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013

[4]

Stephen C. Preston, Alejandro Sarria. One-parameter solutions of the Euler-Arnold equation on the contactomorphism group. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2123-2130. doi: 10.3934/dcds.2015.35.2123

[5]

Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$-Component Camassa-Holm equation with peakons. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1575-1601. doi: 10.3934/dcds.2017065

[6]

José Manuel Palacios. Orbital and asymptotic stability of a train of peakons for the Novikov equation. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2475-2518. doi: 10.3934/dcds.2020372

[7]

Andrey Tsiganov. Integrable Euler top and nonholonomic Chaplygin ball. Journal of Geometric Mechanics, 2011, 3 (3) : 337-362. doi: 10.3934/jgm.2011.3.337

[8]

Yury Neretin. The group of diffeomorphisms of the circle: Reproducing kernels and analogs of spherical functions. Journal of Geometric Mechanics, 2017, 9 (2) : 207-225. doi: 10.3934/jgm.2017009

[9]

Aristophanes Dimakis, Folkert Müller-Hoissen. Bidifferential graded algebras and integrable systems. Conference Publications, 2009, 2009 (Special) : 208-219. doi: 10.3934/proc.2009.2009.208

[10]

Leo T. Butler. A note on integrable mechanical systems on surfaces. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1873-1878. doi: 10.3934/dcds.2014.34.1873

[11]

Aiyong Chen, Xinhui Lu. Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1703-1735. doi: 10.3934/dcds.2020090

[12]

Yongsheng Mi, Chunlai Mu. On a three-Component Camassa-Holm equation with peakons. Kinetic & Related Models, 2014, 7 (2) : 305-339. doi: 10.3934/krm.2014.7.305

[13]

Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194

[14]

Byungsoo Moon. Orbital stability of periodic peakons for the generalized modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021123

[15]

Sonomi Kakizaki, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura. Conserved quantities of the integrable discrete hungry systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 889-899. doi: 10.3934/dcdss.2015.8.889

[16]

Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109

[17]

Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61

[18]

Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001

[19]

Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325

[20]

Boris S. Kruglikov and Vladimir S. Matveev. Vanishing of the entropy pseudonorm for certain integrable systems. Electronic Research Announcements, 2006, 12: 19-28.

2020 Impact Factor: 0.857

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]